2() = 21i,t - i2 (-1+)(ER(i,) - )

That is, \(E_{i}^c(t) \subseteq \Psi _1(\xi ) \cup \Psi _2(\xi )\) . We then use the union bound which yields \(\mathbb {P}(E^c_{i}(t))\le (\Psi _1(\xi )) + (\Psi _2(\xi ))\) , which we can upper bound by known concentration bounds. Following the strategy employed by [1]}, a judicious selection of the free parameter \(\xi \in (0,1)\)  allows us to allocate "weights" on the bounds of \((\Psi _1(\xi ))\)  and \((\Psi _2(\xi ))\)  which then yield \( {2}/{(\xi ^2\Delta _{}^2(i,\gamma ))}\)  and \({h {\frac{\gamma \sigma _i^2}{\gamma \sigma _i^2-2(1-\xi ){i}}}}^{-1}\)  without incurring further residual terms.
